Formklassen

Formklassen, literally “form classes” in several Scandinavian languages, is a concept used in geometry, design theory, and related fields to describe groups of shapes that are considered equivalent under a specified set of transformations. The idea is to classify shapes not by their position or size, but by their intrinsic form when subjected to a chosen transformation group.

Mathematically, Formklassen are the orbits of a set of shapes S under the action of a transformation group G. If s is a shape in S, its Formklasse is the collection {g · s | g ∈ G}. Different choices of G yield different levels of similarity: rigid motions (translations and rotations) produce congruence classes; similarities allow scaling; affine transformations yield broader classes. For example, under rigid motions in the plane, all circles belong to a single Formklasse, while line segments of different lengths belong to different classes unless scaling is also allowed.

Historically, the term appears in Nordic technical and design literature to facilitate comparison and analysis of forms irrespective of placement, orientation, or size. In practice, Formklassen support pattern recognition, shape matching, and morphometric studies, and they inform decision making in industrial design and computer-aided design by focusing on form rather than superficial embedding details.

Variants and related concepts include the use of invariants—properties preserved under G that help identify Formklassen—and broader shape analysis topics such as orbit classification and group-action-based metrics. See also group action, orbit, invariants, pattern recognition, and shape analysis.